Please could some one help me with this trig question?

Question

Solve the equation: 4sin 2x + 3 cos 2x = 2

I'm confused by the 2x bits otherwise I could do it!!

Please help me by pointing me in the right direction!!

(Don't bother posting answers saying do your own homework, I have tried, I cant answer this one question so am asking for help!!)

Answer 1

Emma, this could be a difficult one for you. So, I've solved it for you. Have patience & check every step carefully. You should then be able to understand it. Cheers!

4sin2x + 3cos2x = 2
=> 4*(2 sinx cosx) + 3*(cos^2 x - sin^2 x) = 2
=> 8sinx cosx + 3*[1-sin^2 x - sin^2 x] = 2
=> 8sinx (1 - sin^2 x)^1/2 + 3 - 6sin^2 x = 2
=> 8sinx (1 - sin^2 x)^1/2 = 6sin^2 x - 1

Squaring both sides, we get,

64 sin^2 x (1 - sin^2 x) = 36 sin^4 x - 12 sin^2 x +1

Let us now give another variable, y, the value of x^2. Then, y=x^2.
So, substituting y=sin^2 x in our equation, we get,

64y(1-y) = 36y^2 - 12y +1
=> 64y - 64y^2 = 36y^2 - 12y + 1
=> 100 y^2 - 76y + 1=0

Using the Quadratic Formula to find the solutions of this
equation, we get,

y = [-(-76) +or- (76^2 - 4*100*1)^1/2] / 2*100
=>y = [76 +or- (5776 - 400)^1/2] / 200
=> y = [76 +or- (5376)^1/2] / 200
=> y = [76 +or- 73.32] / 200
=> y = (76 + 73.32) / 200, or, y = (76 - 73.32) / 200
=> y = 0.7466, or, y = 0.0134

Now, resubstituting the actual value of y in the equation, we get,

sin^2 x = 0.7466, or, sin^2 x = 0.0134
=> sin x = +or- 0.864, or, sin x =+or- 0.116 [By taking square root of both sides]
=> x = sin^-1 (+or- 0.864), or, x = sin^-1 (+or- 0.116)
=> x = +59.77 degrees , or, -59.77 degrees, or, 6.66 degrees , or, -6.66 degrees.

Answer 2

try taking the 2 to the other side and change it with sin squared x + cos squared x = 1. Then factorize what is left using the quadratic root thing. Then rearrange to get tan x for both parts and the answers are there.
either
x = tan^-1(1/(-4-sqrt21)
or x = tan^-1(1/(-4+sqrt21)

Answer 3

If the 2x confuses you, replace it with y.

When you have solved for y, remember that y=2x so use this to get at x.

Answer 4

look in your reference book for trigonometric identities. You will find some double angle formulae. This is exactly what 2x means is double the value of the angle x.

Answer 5

cos2x=cos^x-sin^x = 1 - 2sin^x
sin2x=2sinxcosx
so 8sinxcosx + 3 - 6sin^x =2
so 6sin^x - 8sinxcosx -1 = 0
etc
I think because this question is so difficult to solve that you may have copied it down wrongly?

Answer 6

I'm not a big trig fan myself. However, I did find this link. This is probably an exercise in identities.


http://mcraeclan.com/MathHelp/GeometryTrigEquivCos3xEtc.htm

Hope this helps.

Answer 7

NO IT'S 4